Ball-avoiding theorems

Consider a nice hyperbolic dynamical system (singularities not excluded). S tatements about the topological smallness of the subset of orbits, which av oid an open subset of the phase space (for every moment of time, or just fo r a not too small subset of times), play a key role in showing hyperbolicit y or ergodicity of semi-dispersive billiards, especially, of hard-ball syst ems. As well as surveying the characteristic results, called ball avoiding theorems, and giving an idea of the methods of their proofs, their applicat ions are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dim ension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets a re naturally related to repellers extensively studied by physicists. For th e interested reader we also sketch some analytical and rigorous results abo ut repellers and escape times.